The King needs your help Extended

This note arose from comments on the problem: The King needs your help

An ambitious king plans to build several new cities in the wilderness, connected by a network of roads. He expects the annual tax revenues from each city to be numerically equal to the square of the population. Road maintenance will be expensive, though; the annual cost for each road is expected to be numerically equal to the product of the populations of the two cities that road connects. The project is considered viable as long as the tax revenues exceed the cost, regardless of the (positive) populations of the various cities.

If the road network was a loop, what can we say about the potential revenue?
If the road network was a complete graph, what can we say about the potential revenue?
If the road network was a tree, what can we say about the potential revenue?

For further investigations

What kind of road networks could we submit, such that any population size could be supported (revenue neutral)?
What kind of road networks could we submit, such that any non-zero population size would result in a surplus (revenue generating)?
Given any road network, how would you determine if there is a potential population size that would result in a deficit?
Given a fixed population size and road network, how would you distribute them to generate the king the most amount of money? The least amount of money?

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Otto Bretscher I made you the author of this note, so that you can "Edit this note" accordingly, and decide where to take this. If you "subscribe to this thread", you will receive notifications of comments / replies.

Calvin Lin Staff - 6 years, 1 month ago

@Otto Bretscher can you tell me,how to solve infinite sum problems ?

A Former Brilliant Member - 5 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$